Invariants in the paramagnetic resonance spectra of impurity-doped crystals

We show that in cubic crystals with anisotropic impurity centers the sum of squares of the magnetic resonance [electron paramagnetic resonance (EPR)] frequencies is invariant with respect to the magnetic field direction. The connection between such an invariant and the $g$-tensor components of the impurity is derived for different types of centers. The established regularity is confirmed experimentally for the spin-noise spectra of a cubic $\mathrm{Ca}{\mathrm{F}}_2-{\mathrm{Nd}}^{3+}$ crystal. We show how this property of the EPR spectra can be efficiently used for the assignment of paramagnetic centers in cubic crystals.

Figure 1

Schematic of the homodyne-detection setup with a rotating magnetic field of constant strength. The laser beam is split at the input polarizing beam splitter (PBS) of the Mach-Zehnder interferometer into two parts: the signal (probe) and local oscillator (LO). The transmitted and scattered photons of the probe have orthogonal linear polarizations. The polarizer is used to filter out the transmitted part. The passed scattered light and the light of the LO are matched to interfere on the 50:50 nonpolarizing beam splitter (nPBS). The balanced photoreceiver measures the interference signal. The AC part of the signal is analyzed using a spectrum analyzer.

Figure 2

Experimental angular dependence of the SN spectrum of the crystal $\mathrm{Ca}{\mathrm{F}}_2-{\mathrm{Nd}}^{3+}$ (0.1 mol $\%$), obtained by rotation of the applied magnetic field (created by the outer permanent magnet). The dashed curves represent fits of the experimental angular dependencies of the SN resonance frequencies calculated by the relationship $\hslash \omega /\mu B_m=\sqrt{a+bcos\left[2\alpha \right]+csin\left[2\alpha \right]}$. The solid white circles in the upper part of the figure are calculated after the expression $\sqrt{{\sum }_{i=1}^3{[\hslash {\omega }_i/\mu B_m]}^2}$, with the frequencies ${\omega }_i$ extracted from the experimental SN spectra. The solid white line shows the average of the extracted values, and the pale white stripe indicates the confidence interval for the measured quantity. The dotted white line represents the invariant obtained from Eq. (11). As is seen, these two lines coincide well within the experimental error.